function r = hydro(Ti, TIME_STEPS)
  close all;
  fig = figure; 
  load('cm', 'mycmap');
  set(fig, 'Colormap', mycmap);

  PARALLEL = false;
  GPU = false;
  HYDROLIZE = false;

  GRID_SIZE = 128;
  DX = 0.01;
  DT = 0.00001;
  VECTOR_SUBSAMPLING = 16;
  [gridx, gridy] = meshgrid([1:VECTOR_SUBSAMPLING:GRID_SIZE], [1:VECTOR_SUBSAMPLING:GRID_SIZE]);


  VIDEO = false;
  FPS = 30;
  SECONDS_PER_LIGHTYEAR = 15;
  if (VIDEO)
    vt = VideoWriter('v_t.avi', 'Archival');
    open(vt);
  end

  ZERO_HYDRO = false;



  A = sqrt(3)*pi/18 - log(3)/2;

  vx = zeros(GRID_SIZE, GRID_SIZE);
  vy = zeros(GRID_SIZE, GRID_SIZE);
  px = zeros(GRID_SIZE, GRID_SIZE);
  py = zeros(GRID_SIZE, GRID_SIZE);
  Pi = zeros(GRID_SIZE, GRID_SIZE);
  B  = zeros(GRID_SIZE, GRID_SIZE);
  T  = zeros(GRID_SIZE, GRID_SIZE);
  % shortcuts
  wx = zeros(GRID_SIZE, GRID_SIZE); % 1 - vx^2
  wy = zeros(GRID_SIZE, GRID_SIZE); % 1 - vy^2
  G = zeros(GRID_SIZE, GRID_SIZE); % sqrt(-1 + wx + wy)
  
  dtT = zeros(GRID_SIZE, GRID_SIZE); 
  dydtT = zeros(GRID_SIZE, GRID_SIZE); 
  dxdxT = zeros(GRID_SIZE, GRID_SIZE); 
  dxdyT = zeros(GRID_SIZE, GRID_SIZE); 
  dydyT = zeros(GRID_SIZE, GRID_SIZE); 
  dtdtvx = zeros(GRID_SIZE, GRID_SIZE); 
  dtdtvy = zeros(GRID_SIZE, GRID_SIZE); 
  dtpx = zeros(GRID_SIZE, GRID_SIZE); 
  dtpy = zeros(GRID_SIZE, GRID_SIZE); 

  % derviatives
  dxvx = zeros(GRID_SIZE, GRID_SIZE);
  dxvy = zeros(GRID_SIZE, GRID_SIZE);
  dxpx = zeros(GRID_SIZE, GRID_SIZE);
  dxpy = zeros(GRID_SIZE, GRID_SIZE);
  dxPi = zeros(GRID_SIZE, GRID_SIZE);
  dxB  = zeros(GRID_SIZE, GRID_SIZE);
  dxT  = zeros(GRID_SIZE, GRID_SIZE);
  dyvx = zeros(GRID_SIZE, GRID_SIZE);
  dyvy = zeros(GRID_SIZE, GRID_SIZE);
  dypx = zeros(GRID_SIZE, GRID_SIZE);
  dypy = zeros(GRID_SIZE, GRID_SIZE);
  dyPi = zeros(GRID_SIZE, GRID_SIZE);
  dyB  = zeros(GRID_SIZE, GRID_SIZE);
  dyT  = zeros(GRID_SIZE, GRID_SIZE);
  dxdxvx = zeros(GRID_SIZE, GRID_SIZE);
  dxdyvx = zeros(GRID_SIZE, GRID_SIZE);
  dxdxvy = zeros(GRID_SIZE, GRID_SIZE);
  dxdyvy = zeros(GRID_SIZE, GRID_SIZE);
  dydyvx = zeros(GRID_SIZE, GRID_SIZE);
  dydyvy = zeros(GRID_SIZE, GRID_SIZE);


  % ********************************************
  % parallel computing
  % ********************************************

  if (PARALLEL)
    px = distributed(px);
    py = distributed(py);
    dtT = distributed(dtT);
    dxvx = distributed(dxvx);
    dyvx = distributed(dyvx);
    dxvy = distributed(dxvy);
    dyvy = distributed(dyvy);
    dxT = distributed(dxvx);
    dyT = distributed(dyvx);
    
    matlabpool;
  end

  sumT = 0;
  % initial conditions for vx vy come here
  % vx
  % vy
  % T
  TEMP_OFFSET = 1000;
  cx = 0.5 * GRID_SIZE + 0.5;
  cy = 0.5 * GRID_SIZE + 0.5;
  for x = 1:GRID_SIZE
    for y = 1:GRID_SIZE
       T(x, y) = 0.004 * exp(-((x-cx)^2 + (y-cy)^2)/72);
      % T(x,y) = 0.2*sin(2*pi*x/GRID_SIZE);
    end
  end
  % T(:,:) = Ti(:,:);
  T = T + TEMP_OFFSET;
  for x = 1:GRID_SIZE
     for y = 1:GRID_SIZE
       %T(x,y) = T(x,y) +  0.01 * rand();
       %vx(x,y) = 0.001 * rand();
       %vy(x,y) = 0.001 * rand();
     end
  end

  vx = 0.000000 * ones(GRID_SIZE, GRID_SIZE);;
  vy = 0.2 * ones(GRID_SIZE, GRID_SIZE);;
  
  % ****************************************************************
  %  hydrolize the initial conditions
  % ****************************************************************
  if (HYDROLIZE)
    level = 4;
    k = zeros(GRID_SIZE+1, GRID_SIZE+1);  
    [cx, cy] = meshgrid(0:GRID_SIZE-1,0:GRID_SIZE-1);
    [kx, ky] = meshgrid(-GRID_SIZE/2:GRID_SIZE/2);
    for kx = -GRID_SIZE/level:GRID_SIZE/level
      for ky = -GRID_SIZE/level:GRID_SIZE/level
        ft = T .* exp(-2 * pi * i * ( kx * cx + ky * cy ) / GRID_SIZE);
        k(kx+GRID_SIZE/2+1, ky+GRID_SIZE/2+1) = sum(sum(ft));
      end
    end
    T = zeros(GRID_SIZE:GRID_SIZE);
    for kx = -GRID_SIZE/level:GRID_SIZE/level
      for ky = -GRID_SIZE/level:GRID_SIZE/level
        T = T + k(kx+GRID_SIZE/2+1,ky+GRID_SIZE/2+1) * exp(2 * pi * i * ( kx * cx + ky * cy ) / GRID_SIZE);
        % vx = vx + 20*(rand()-0.5)*exp(2 * pi * i * ( kx * cx + ky * cy ) / GRID_SIZE);
        % vy = vy + 20*(rand()-0.5)*exp(2 * pi * i * ( kx * cx + ky * cy ) / GRID_SIZE);

      end
    end
    T = real(T / GRID_SIZE / GRID_SIZE);
    vx = real(vx / GRID_SIZE / GRID_SIZE);
    vy = real(vy / GRID_SIZE / GRID_SIZE);
  end



  % ****************************************************************
  %  compute derivative kernels
  % ****************************************************************
  % can upgrade on this -- trefethen pg 6

  d1 = zeros(GRID_SIZE, GRID_SIZE);
  d2 = zeros(GRID_SIZE, GRID_SIZE);
  for x = 1:GRID_SIZE
    for y=1:GRID_SIZE
      s1 = 0;
      s2 = 0;
      for k = -GRID_SIZE/2 : GRID_SIZE/2
        s1 = s1 + 2. * pi * i / GRID_SIZE / GRID_SIZE * k * exp(2 * pi * i * k * (x - y) / GRID_SIZE);
        s2 = s2 + (2. * pi * i / GRID_SIZE * k)^2 / GRID_SIZE * exp(2 * pi * i * k * (x - y) / GRID_SIZE);
      end
      d1(x,y) = real(s1)/DX;
      d2(x,y) = real(s2)/DX/DX;
    end
  end
  h = 2*pi/GRID_SIZE;
  col1 = [0 .5*(-1).^(1:GRID_SIZE-1).*cot((1:GRID_SIZE-1)*h/2)]  * h/DX;
  max(max(d1 - toeplitz(col1, col1([1 GRID_SIZE:-1:2]))))
  col2 = [(-pi^2/(3*h*h)-1/6) .5*(-1).^(0:GRID_SIZE-2).*(csc((1:GRID_SIZE-1)*h/2).^2)]  * h*h/(DX*DX);
  % d2 = toeplitz(col2);
  

  %dxT = dxdx(T);
  %dxT2 = dxdx2(T);
  % image(imagesc(dxT));
  %figure;
  %image(imagesc(dxT2));
  %image((dxT) * 128  + 128); 
  % image((dxT) * 128 + 128); 
  % truesize(fig);
  %fig=figure;
  %truesize(fig);
  %keyboard();
 % image((T-TEMP_OFFSET)*256+128);

  %return

  % ****************************************************************
  % compute the higher order initial conditions from ideal hydro:
  % ****************************************************************

  tic;

  wx = 1 - vx .* vx;
  wy = 1 - vy .* vy;
  G = sqrt(-1 + wx + wy);


  % compute the gradients of initial data
  dxvx = dx(vx);
  dyvx = dy(vx);
  dxvy = dx(vy);
  dyvy = dy(vy);
  dxT = dx(T);
  dyT = dy(T);
  dxdxvx = dxdx(vx);
  dxdyvx = dx(dy(vx));
  dydyvx = dydy(vx);

  dxdxvy = dxdx(vy);
  dxdyvy = dx(dy(vy));
  dydyvy = dydy(vy);

  dxdxT = dxdx(T);
  dxdyT = dx(dy(T));
  dydyT = dydy(T);

  px = (1./(T.*(-2 + vx.^2 + vy.^2))).*(vx.^3.*(vy.*dyT + T.*dyvy) + 2.*vx.^4.* dxT - (-2 + vy.^2).*(T.*vy.* dyvx - (-1 + vy.^2).* dxT) + vx.^2.*((-T).*vy.* dyvx + (-4 + 3.*vy.^2).* dxT) + vx.*((-1 + vy.^2).*(vy.* dyT + T.*dyvy) + T.*dxvx));
  py = ((-1 + vx.^2 + vy.^2).*(-2 + vx.^2 + 2.*vy.^2).*dyT + vx.*vy.*(-1 + vx.^2 + vy.^2).* dxT + T.*(vy.*(dyvy + (-1 + vx.^2 + vy.^2).* dxvx) - vx.*(-2 + vx.^2 + vy.^2).* dxvy))./(T.*(-2 + vx.^2 + vy.^2));


  dxpx = dx(px);
  dypx = dy(px);
  dxpy = dx(py);
  dypy = dy(py);




  dtdtvx = (1./(4.*pi.*T.*(-1 + vx.^2 + vy.^2).^3.*(vx.^2 + vy.^2))).*(sqrt(1 - vx.^2 - vy.^2).*(16.*pi.^2.*T.*vx.*vy.*dyT - 64.*pi.^2.*T.*vx.^3.*vy.*dyT + 96.*pi.^2.*T.*vx.^5.*vy.*dyT - 64.*pi.^2.*T.*vx.^7.*vy.*dyT + 16.*pi.^2.*T.*vx.^9.*vy.*dyT - 64.*pi.^2.*T.*vx.*vy.^3.*dyT + 192.*pi.^2.*T.*vx.^3.*vy.^3.*dyT - 192.*pi.^2.*T.*vx.^5.*vy.^3.*dyT + 64.*pi.^2.*T.*vx.^7.*vy.^3.*dyT + 96.*pi.^2.*T.*vx.*vy.^5.*dyT - 192.*pi.^2.*T.*vx.^3.*vy.^5.*dyT + 96.*pi.^2.*T.*vx.^5.*vy.^5.*dyT - 64.*pi.^2.*T.*vx.*vy.^7.*dyT + 64.*pi.^2.*T.*vx.^3.*vy.^7.*dyT + 16.*pi.^2.*T.*vx.*vy.^9.*dyT - 16.*pi.^2.*T.^2.*vy.*dyvx + 56.*pi.^2.*T.^2.*vx.^2.*vy.*dyvx - 72.*pi.^2.*T.^2.*vx.^4.*vy.*dyvx + 40.*pi.^2.*T.^2.*vx.^6.*vy.*dyvx - 8.*pi.^2.*T.^2.*vx.^8.*vy.*dyvx + 48.*pi.^2.*T.^2.*vy.^3.*dyvx - 120.*pi.^2.*T.^2.*vx.^2.*vy.^3.*dyvx + 96.*pi.^2.*T.^2.*vx.^4.*vy.^3.*dyvx - 24.*pi.^2.*T.^2.*vx.^6.*vy.^3.*dyvx - 48.*pi.^2.*T.^2.*vy.^5.*dyvx + 72.*pi.^2.*T.^2.*vx.^2.*vy.^5.*dyvx - 24.*pi.^2.*T.^2.*vx.^4.*vy.^5.*dyvx + 16.*pi.^2.*T.^2.*vy.^7.*dyvx - 8.*pi.^2.*T.^2.*vx.^2.*vy.^7.*dyvx + vy.*dyvx.^3 - 2.*vx.^2.*vy.*dyvx.^3 + vx.^4.*vy.*dyvx.^3 - vy.^3.*dyvx.^3 + 2.*vx.^2.*vy.^3.*dyvx.^3 - vx.^4.*vy.^3.*dyvx.^3 + 8.*pi.^2.*T.^2.*vx.*dyvy - 32.*pi.^2.*T.^2.*vx.^3.*dyvy + 48.*pi.^2.*T.^2.*vx.^5.*dyvy - 32.*pi.^2.*T.^2.*vx.^7.*dyvy + 8.*pi.^2.*T.^2.*vx.^9.*dyvy - 24.*pi.^2.*T.^2.*vx.*vy.^2.*dyvy + 72.*pi.^2.*T.^2.*vx.^3.*vy.^2.*dyvy - 72.*pi.^2.*T.^2.*vx.^5.*vy.^2.*dyvy + 24.*pi.^2.*T.^2.*vx.^7.*vy.^2.*dyvy + 24.*pi.^2.*T.^2.*vx.*vy.^4.*dyvy - 48.*pi.^2.*T.^2.*vx.^3.*vy.^4.*dyvy + 24.*pi.^2.*T.^2.*vx.^5.*vy.^4.*dyvy - 8.*pi.^2.*T.^2.*vx.*vy.^6.*dyvy + 8.*pi.^2.*T.^2.*vx.^3.*vy.^6.*dyvy - vx.*dyvx.^2.*dyvy + 2.*vx.^3.*dyvx.^2.*dyvy - vx.^5.*dyvx.^2.*dyvy + 3.*vx.*vy.^2.*dyvx.^2.*dyvy - 6.*vx.^3.*vy.^2.*dyvx.^2.*dyvy + 3.*vx.^5.*vy.^2.*dyvx.^2.*dyvy + vy.*dyvx.*dyvy.^2 - 5.*vx.^2.*vy.*dyvx.*dyvy.^2 + 7.*vx.^4.*vy.*dyvx.*dyvy.^2 - 3.*vx.^6.*vy.*dyvx.*dyvy.^2 - vx.*dyvy.^3 + 3.*vx.^3.*dyvy.^3 - 3.*vx.^5.*dyvy.^3 + vx.^7.*dyvy.^3 - 16.*pi.^2.*T.*dxT + 80.*pi.^2.*T.*vx.^2.*dxT - 160.*pi.^2.*T.*vx.^4.*dxT + 160.*pi.^2.*T.*vx.^6.*dxT - 80.*pi.^2.*T.*vx.^8.*dxT + 16.*pi.^2.*T.*vx.^10.*dxT + 64.*pi.^2.*T.*vy.^2.*dxT - 256.*pi.^2.*T.*vx.^2.*vy.^2.*dxT + 384.*pi.^2.*T.*vx.^4.*vy.^2.*dxT - 256.*pi.^2.*T.*vx.^6.*vy.^2.*dxT + 64.*pi.^2.*T.*vx.^8.*vy.^2.*dxT - 96.*pi.^2.*T.*vy.^4.*dxT + 288.*pi.^2.*T.*vx.^2.*vy.^4.*dxT - 288.*pi.^2.*T.*vx.^4.*vy.^4.*dxT + 96.*pi.^2.*T.*vx.^6.*vy.^4.*dxT + 64.*pi.^2.*T.*vy.^6.*dxT - 128.*pi.^2.*T.*vx.^2.*vy.^6.*dxT + 64.*pi.^2.*T.*vx.^4.*vy.^6.*dxT - 16.*pi.^2.*T.*vy.^8.*dxT + 16.*pi.^2.*T.*vx.^2.*vy.^8.*dxT - 8.*pi.^2.*T.^2.*vx.*dxvx + 24.*pi.^2.*T.^2.*vx.^3.*dxvx - 24.*pi.^2.*T.^2.*vx.^5.*dxvx + 8.*pi.^2.*T.^2.*vx.^7.*dxvx + 16.*pi.^2.*T.^2.*vx.*vy.^2.*dxvx - 24.*pi.^2.*T.^2.*vx.^3.*vy.^2.*dxvx + 8.*pi.^2.*T.^2.*vx.^7.*vy.^2.*dxvx - 24.*pi.^2.*T.^2.*vx.^3.*vy.^4.*dxvx + 24.*pi.^2.*T.^2.*vx.^5.*vy.^4.*dxvx - 16.*pi.^2.*T.^2.*vx.*vy.^6.*dxvx + 24.*pi.^2.*T.^2.*vx.^3.*vy.^6.*dxvx + 8.*pi.^2.*T.^2.*vx.*vy.^8.*dxvx + vx.*dyvx.^2.*dxvx - vx.^3.*dyvx.^2.*dxvx + 2.*vx.*vy.^2.*dyvx.^2.*dxvx - 2.*vx.^3.*vy.^2.*dyvx.^2.*dxvx - 3.*vx.*vy.^4.*dyvx.^2.*dxvx + 3.*vx.^3.*vy.^4.*dyvx.^2.*dxvx - 2.*vy.*dyvx.*dyvy.*dxvx + 4.*vx.^2.*vy.*dyvx.*dyvy.*dxvx - 2.*vx.^4.*vy.*dyvx.*dyvy.*dxvx + 2.*vy.^3.*dyvx.*dyvy.*dxvx + 4.*vx.^2.*vy.^3.*dyvx.*dyvy.*dxvx - 6.*vx.^4.*vy.^3.*dyvx.*dyvy.*dxvx + 3.*vx.*dyvy.^2.*dxvx - 6.*vx.^3.*dyvy.^2.*dxvx + 3.*vx.^5.*dyvy.^2.*dxvx - vx.*vy.^2.*dyvy.^2.*dxvx - 2.*vx.^3.*vy.^2.*dyvy.^2.*dxvx + 3.*vx.^5.*vy.^2.*dyvy.^2.*dxvx + vy.*dyvx.*dxvx.^2 + vx.^2.*vy.*dyvx.*dxvx.^2 - 2.*vy.^3.*dyvx.*dxvx.^2 + 2.*vx.^2.*vy.^3.*dyvx.*dxvx.^2 + vy.^5.*dyvx.*dxvx.^2 - 3.*vx.^2.*vy.^5.*dyvx.*dxvx.^2 - 3.*vx.*dyvy.*dxvx.^2 + 3.*vx.^3.*dyvy.*dxvx.^2 + 2.*vx.*vy.^2.*dyvy.*dxvx.^2 + 2.*vx.^3.*vy.^2.*dyvy.*dxvx.^2 + vx.*vy.^4.*dyvy.*dxvx.^2 + 3.*vx.^3.*vy.^4.*dyvy.*dxvx.^2 + vx.*dxvx.^3 - vx.*vy.^2.*dxvx.^3 - vx.*vy.^4.*dxvx.^3 + vx.*vy.^6.*dxvx.^3 + 8.*pi.^2.*T.^2.*vx.^2.*vy.*dxvy - 24.*pi.^2.*T.^2.*vx.^4.*vy.*dxvy + 24.*pi.^2.*T.^2.*vx.^6.*vy.*dxvy - 8.*pi.^2.*T.^2.*vx.^8.*vy.*dxvy - 24.*pi.^2.*T.^2.*vx.^2.*vy.^3.*dxvy + 48.*pi.^2.*T.^2.*vx.^4.*vy.^3.*dxvy - 24.*pi.^2.*T.^2.*vx.^6.*vy.^3.*dxvy + 24.*pi.^2.*T.^2.*vx.^2.*vy.^5.*dxvy - 24.*pi.^2.*T.^2.*vx.^4.*vy.^5.*dxvy - 8.*pi.^2.*T.^2.*vx.^2.*vy.^7.*dxvy + 3.*vy.*dyvx.^2.*dxvy - 6.*vx.^2.*vy.*dyvx.^2.*dxvy + 3.*vx.^4.*vy.*dyvx.^2.*dxvy - 3.*vy.^3.*dyvx.^2.*dxvy + 6.*vx.^2.*vy.^3.*dyvx.^2.*dxvy - 3.*vx.^4.*vy.^3.*dyvx.^2.*dxvy - 2.*vx.*dyvx.*dyvy.*dxvy + 4.*vx.^3.*dyvx.*dyvy.*dxvy - 2.*vx.^5.*dyvx.*dyvy.*dxvy + 6.*vx.*vy.^2.*dyvx.*dyvy.*dxvy - 12.*vx.^3.*vy.^2.*dyvx.*dyvy.*dxvy + 6.*vx.^5.*vy.^2.*dyvx.*dyvy.*dxvy + vy.*dyvy.^2.*dxvy - 5.*vx.^2.*vy.*dyvy.^2.*dxvy + 7.*vx.^4.*vy.*dyvy.^2.*dxvy - 3.*vx.^6.*vy.*dyvy.^2.*dxvy + 2.*vx.*dyvx.*dxvx.*dxvy - 2.*vx.^3.*dyvx.*dxvx.*dxvy + 4.*vx.*vy.^2.*dyvx.*dxvx.*dxvy - 4.*vx.^3.*vy.^2.*dyvx.*dxvx.*dxvy - 6.*vx.*vy.^4.*dyvx.*dxvx.*dxvy + 6.*vx.^3.*vy.^4.*dyvx.*dxvx.*dxvy - 2.*vy.*dyvy.*dxvx.*dxvy + 4.*vx.^2.*vy.*dyvy.*dxvx.*dxvy - 2.*vx.^4.*vy.*dyvy.*dxvx.*dxvy + 2.*vy.^3.*dyvy.*dxvx.*dxvy + 4.*vx.^2.*vy.^3.*dyvy.*dxvx.*dxvy - 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4.*vx.^3.*vy.^2.*dxvx.*px.*py - 4.*vx.*vy.^4.*dxvx.*px.*py + 2.*vx.^3.*vy.^4.*dxvx.*px.*py + 2.*vx.*vy.^6.*dxvx.*px.*py + 2.*vx.^2.*vy.*dxvy.*px.*py + 2.*vx.^4.*vy.*dxvy.*px.*py + 2.*vy.^3.*dxvy.*px.*py - 2.*vx.^4.*vy.^3.*dxvy.*px.*py - 2.*vy.^5.*dxvy.*px.*py - 2.*vx.^2.*vy.^5.*dxvy.*px.*py + vx.^4.*px.^2.*py + 2.*vx.^2.*vy.^2.*px.^2.*py - vx.^4.*vy.^2.*px.^2.*py + vy.^4.*px.^2.*py - 2.*vx.^2.*vy.^4.*px.^2.*py - vy.^6.*px.^2.*py + 3.*vx.^3.*dyvx.*py.^2 - 3.*vx.^5.*dyvx.*py.^2 + 3.*vx.*vy.^2.*dyvx.*py.^2 - 4.*vx.^3.*vy.^2.*dyvx.*py.^2 + vx.^5.*vy.^2.*dyvx.*py.^2 - vx.*vy.^4.*dyvx.*py.^2 + vx.^3.*vy.^4.*dyvx.*py.^2 + 3.*vx.^2.*vy.*dyvy.*py.^2 - 2.*vx.^4.*vy.*dyvy.*py.^2 - vx.^6.*vy.*dyvy.*py.^2 + 3.*vy.^3.*dyvy.*py.^2 - 2.*vx.^2.*vy.^3.*dyvy.*py.^2 - vx.^4.*vy.^3.*dyvy.*py.^2 - 3.*vx.^2.*vy.*dxvx.*py.^2 + 5.*vx.^4.*vy.*dxvx.*py.^2 - 3.*vy.^3.*dxvx.*py.^2 + 8.*vx.^2.*vy.^3.*dxvx.*py.^2 - vx.^4.*vy.^3.*dxvx.*py.^2 + 3.*vy.^5.*dxvx.*py.^2 - vx.^2.*vy.^5.*dxvx.*py.^2 + 3.*vx.^3.*dxvy.*py.^2 - 3.*vx.^5.*dxvy.*py.^2 + 3.*vx.*vy.^2.*dxvy.*py.^2 - 4.*vx.^3.*vy.^2.*dxvy.*py.^2 + vx.^5.*vy.^2.*dxvy.*py.^2 - vx.*vy.^4.*dxvy.*py.^2 + vx.^3.*vy.^4.*dxvy.*py.^2 + 2.*vx.^5.*vy.*px.*py.^2 + 4.*vx.^3.*vy.^3.*px.*py.^2 + 2.*vx.*vy.^5.*px.*py.^2 + vx.^4.*py.^3 - vx.^6.*py.^3 + 2.*vx.^2.*vy.^2.*py.^3 - 2.*vx.^4.*vy.^2.*py.^3 + vy.^4.*py.^3 - vx.^2.*vy.^4.*py.^3)) + (1./(2.*T.*(-1 + vx.^2 + vy.^2).*(vx.^2 + vy.^2))).*(-4.*vx.*vy.*dyT.*dyvx + 4.*vx.^3.*vy.*dyT.*dyvx + 8.*vx.*vy.^3.*dyT.*dyvx - 4.*vx.^3.*vy.^3.*dyT.*dyvx - 4.*vx.*vy.^5.*dyT.*dyvx - T.*vy.*dyvx.^2 - T.*vx.^2.*vy.*dyvx.^2 + T.*vy.^3.*dyvx.^2 + T.*vx.^2.*vy.^3.*dyvx.^2 + 4.*dyT.*dyvy - 8.*vx.^2.*dyT.*dyvy + 4.*vx.^4.*dyT.*dyvy - 8.*vy.^2.*dyT.*dyvy + 4.*vx.^2.*vy.^2.*dyT.*dyvy + 4.*vx.^4.*vy.^2.*dyT.*dyvy + 4.*vy.^4.*dyT.*dyvy + 4.*vx.^2.*vy.^4.*dyT.*dyvy + 2.*T.*vx.*dyvx.*dyvy - 2.*T.*vx.^3.*dyvx.*dyvy + 2.*T.*vx.*vy.^2.*dyvx.*dyvy - 2.*T.*vx.^3.*vy.^2.*dyvx.*dyvy + 5.*T.*vy.*dyvy.^2 - 6.*T.*vx.^2.*vy.*dyvy.^2 + T.*vx.^4.*vy.*dyvy.^2 + 2.*T.*dydyvy - 4.*T.*vx.^2.*dydyvy + 2.*T.*vx.^4.*dydyvy - 2.*T.*vy.^2.*dydyvy + 2.*T.*vx.^2.*vy.^2.*dydyvy + 4.*dyvx.*dxT - 8.*vx.^2.*dyvx.*dxT + 4.*vx.^4.*dyvx.*dxT - 8.*vy.^2.*dyvx.*dxT + 12.*vx.^2.*vy.^2.*dyvx.*dxT - 4.*vx.^4.*vy.^2.*dyvx.*dxT + 4.*vy.^4.*dyvx.*dxT - 4.*vx.^2.*vy.^4.*dyvx.*dxT + 4.*vx.*vy.*dyvy.*dxT - 8.*vx.^3.*vy.*dyvy.*dxT + 4.*vx.^5.*vy.*dyvy.*dxT - 4.*vx.*vy.^3.*dyvy.*dxT + 4.*vx.^3.*vy.^3.*dyvy.*dxT - 4.*dyT.*dxvx + 4.*vx.^2.*dyT.*dxvx + 12.*vy.^2.*dyT.*dxvx - 8.*vx.^2.*vy.^2.*dyT.*dxvx - 12.*vy.^4.*dyT.*dxvx + 4.*vx.^2.*vy.^4.*dyT.*dxvx + 4.*vy.^6.*dyT.*dxvx - 2.*T.*vx.*dyvx.*dxvx + 4.*T.*vx.*vy.^2.*dyvx.*dxvx - 2.*T.*vx.*vy.^4.*dyvx.*dxvx + 2.*T.*vy.*dyvy.*dxvx + 2.*T.*vx.^2.*vy.*dyvy.*dxvx - 2.*T.*vy.^3.*dyvy.*dxvx + 2.*T.*vx.^2.*vy.^3.*dyvy.*dxvx + 4.*vx.*vy.*dxT.*dxvx - 4.*vx.^3.*vy.*dxT.*dxvx - 8.*vx.*vy.^3.*dxT.*dxvx + 4.*vx.^3.*vy.^3.*dxT.*dxvx + 4.*vx.*vy.^5.*dxT.*dxvx + T.*vy.*dxvx.^2 - 2.*T.*vy.^3.*dxvx.^2 + T.*vy.^5.*dxvx.^2 - 4.*vx.*vy.*dyT.*dxvy + 4.*vx.^3.*vy.*dyT.*dxvy + 8.*vx.*vy.^3.*dyT.*dxvy - 4.*vx.^3.*vy.^3.*dyT.*dxvy - 4.*vx.*vy.^5.*dyT.*dxvy + 2.*T.*vy.*dyvx.*dxvy + 2.*T.*vx.^2.*vy.*dyvx.*dxvy - 2.*T.*vy.^3.*dyvx.*dxvy + 2.*T.*vx.^2.*vy.^3.*dyvx.*dxvy + 2.*T.*vx.*dyvy.*dxvy - 2.*T.*vx.^3.*dyvy.*dxvy + 10.*T.*vx.*vy.^2.*dyvy.*dxvy - 2.*T.*vx.^3.*vy.^2.*dyvy.*dxvy + 4.*dxT.*dxvy - 8.*vx.^2.*dxT.*dxvy + 4.*vx.^4.*dxT.*dxvy - 8.*vy.^2.*dxT.*dxvy + 12.*vx.^2.*vy.^2.*dxT.*dxvy - 4.*vx.^4.*vy.^2.*dxT.*dxvy + 4.*vy.^4.*dxT.*dxvy - 4.*vx.^2.*vy.^4.*dxT.*dxvy + 6.*T.*vx.*dxvx.*dxvy - 4.*T.*vx.*vy.^2.*dxvx.*dxvy - 2.*T.*vx.*vy.^4.*dxvx.*dxvy + 3.*T.*vy.*dxvy.^2 + 3.*T.*vx.^2.*vy.*dxvy.^2 - 3.*T.*vy.^3.*dxvy.^2 + T.*vx.^2.*vy.^3.*dxvy.^2 + 4.*T.*vx.*vy.*dxdyvy - 4.*T.*vx.^3.*vy.*dxdyvy - 4.*T.*vx.*vy.^3.*dxdyvy + 2.*T.*dxdxvy - 2.*T.*vx.^2.*dxdxvy - 4.*T.*vy.^2.*dxdxvy + 2.*T.*vx.^2.*vy.^2.*dxdxvy + 2.*T.*vy.^4.*dxdxvy - 4.*vx.*dyT.*px + 4.*vx.^3.*dyT.*px + 8.*vx.*vy.^2.*dyT.*px - 4.*vx.^3.*vy.^2.*dyT.*px - 4.*vx.*vy.^4.*dyT.*px - 2.*T.*vx.^2.*dyvx.*px - 2.*T.*vy.^2.*dyvx.*px + 2.*T.*vx.^2.*vy.^2.*dyvx.*px + 2.*T.*vy.^4.*dyvx.*px + 8.*T.*vx.*vy.*dyvy.*px - 4.*T.*vx.^3.*vy.*dyvy.*px - 4.*T.*vx.*vy.^3.*dyvy.*px + 4.*vy.*dxT.*px - 4.*vx.^2.*vy.*dxT.*px - 8.*vy.^3.*dxT.*px + 4.*vx.^2.*vy.^3.*dxT.*px + 4.*vy.^5.*dxT.*px + 4.*T.*dxvy.*px + 2.*T.*vx.^2.*dxvy.*px - 6.*T.*vy.^2.*dxvy.*px + 2.*T.*vx.^2.*vy.^2.*dxvy.*px + 2.*T.*vy.^4.*dxvy.*px - T.*vx.^2.*vy.*px.^2 - T.*vy.^3.*px.^2 + T.*vx.^2.*vy.^3.*px.^2 + T.*vy.^5.*px.^2 + 4.*vy.*dyT.*py - 12.*vx.^2.*vy.*dyT.*py + 8.*vx.^4.*vy.*dyT.*py - 8.*vy.^3.*dyT.*py + 12.*vx.^2.*vy.^3.*dyT.*py + 4.*vy.^5.*dyT.*py + 4.*T.*vx.*vy.*dyvx.*py + 4.*T.*dyvy.*py - 6.*T.*vx.^2.*dyvy.*py + 2.*T.*vx.^4.*dyvy.*py + 6.*T.*vy.^2.*dyvy.*py + 2.*T.*vx.^2.*vy.^2.*dyvy.*py + 4.*vx.*dxT.*py - 8.*vx.^3.*dxT.*py + 4.*vx.^5.*dxT.*py - 4.*vx.*vy.^2.*dxT.*py + 4.*vx.^3.*vy.^2.*dxT.*py + 6.*T.*vx.^2.*dxvx.*py + 2.*T.*vy.^2.*dxvx.*py - 2.*T.*vx.^2.*vy.^2.*dxvx.*py - 2.*T.*vy.^4.*dxvx.*py + 12.*T.*vx.*vy.*dxvy.*py + 4.*T.*vx.*px.*py + 2.*T.*vx.^3.*px.*py + 2.*T.*vx.*vy.^2.*px.*py - 2.*T.*vx.^3.*vy.^2.*px.*py - 2.*T.*vx.*vy.^4.*px.*py + 4.*T.*vy.*py.^2 + T.*vx.^2.*vy.*py.^2 + T.*vx.^4.*vy.*py.^2 + T.*vy.^3.*py.^2 + T.*vx.^2.*vy.^3.*py.^2 + 4.*T.*vy.*dypy - 4.*T.*vx.^2.*vy.*dypy - 4.*T.*vy.^3.*dypy + 4.*T.*vx.*dxpy - 4.*T.*vx.^3.*dxpy - 4.*T.*vx.*vy.^2.*dxpy)


  % we don't really need these two
  dtpx = dtdtvx;
  dtpy = dtdtvy;

















% .*.*.*.*.*.*.*.*.*.*.*.*.*.*.*.*.*.*.*.*.*.*.*.*.*.*.*.*.*.*.*.*.*.*.*.*.*.*.*.*.*.*.*.*



  Pi = (1./(9.*(1 - vx.^2 - vy.^2).^(9./2))).*(pi.*T.*(-8.*pi.*T.*(-1 + vx.^2 + vy.^2).^2.*((-1 + vx.^2).*(-1 + vy.^2).*dyvx + vx.^2.*(-1 + vy.^2).*dxvy - (-1 + vy.^2).*(dxvy + vy.*px) - vx.^3.*(vy.*dyvy + py) + vx.*((-vy.^3).*dxvx + vy.*(dyvy + dxvx) + py)) + 3.*sqrt(1 - vx.^2 - vy.^2).*(2.*(1 + A).*vx.^6.*(vy.*dxdyvy + dxpy) - vx.^4.*((-dyvy).*dxvy - 3.*A.*dyvy.*dxvy + 8.*dxvx.*dxvy + 8.*A.*dxvx.*dxvy + 2.*(1 + A).*vy.^3.*(dydyvx - dxdxvx) + 5.*px.*py + 3.*A.*px.*py + dyvx.*((-1 + A + 6.*(1 + A).*vy.^2).*dyvy - 4.*(1 + A).*(-1 + vy.^2).*dxvx + (5 + 7.*A).*vy.*py) + vy.*(-2.*(1 + A).*dydyvx + 2.*(1 + A).*dxdyvy + 2.*dxdxvx + 2.*A.*dxdxvx + 5.*dyvy.*px + 7.*A.*dyvy.*px + 9.*dxvy.*py + 7.*A.*dxvy.*py) - 2.*dypx - 2.*A.*dypx + 2.*vy.^2.*(5.*(1 + A).*dyvy.*dxvy - 4.*(1 + A).*dxvx.*dxvy + 2.*A.*px.*py + dypx + A.*dypx) + 2.*dxpy + 2.*A.*dxpy) + vx.^2.*(-2.*dyvy.*dxvy - 6.*A.*dyvy.*dxvy + 7.*dxvx.*dxvy + 9.*A.*dxvx.*dxvy - 2.*(1 + A).*vy.^5.*(dydyvx + dxdyvy - dxdxvx) + px.*py - A.*px.*py + dyvx.*((2.*(-1 + A) - (1 + 3.*A).*vy.^2 + 8.*(1 + A).*vy.^4).*dyvy - (-1 + vy.^2).*(3 + A + 10.*(1 + A).*vy.^2).*dxvx + vy.*(2.*(-1 + A) + (7 + 5.*A).*vy.^2).*py) - 4.*dypx - 4.*A.*dypx - 2.*dxpy - 2.*A.*dxpy + vy.^4.*(4.*(1 + A).*dyvy.*dxvy - 2.*(3.*(1 + A).*dxvx.*dxvy + 2.*A.*px.*py + (1 + A).*dxpy)) + vy.^2.*((7 + 9.*A).*dyvy.*dxvy - (1 + 3.*A).*dxvx.*dxvy + 4.*((1 + 3.*A).*px.*py + (1 + A).*(dypx + dxpy))) + vy.*(-4.*(1 + A).*dydyvx - 2.*(1 + A).*dxdyvy + 2.*dxdxvx + 2.*A.*dxdxvx - 2.*dyvy.*px + 2.*A.*dyvy.*px + 13.*dxvx.*px + 11.*A.*dxvx.*px + 4.*dxvy.*py - 2.*dtdtvx - 2.*A.*dtdtvx) + vy.^3.*(6.*(1 + A).*dydyvx + 4.*(1 + A).*dxdyvy - 4.*dxdxvx - 4.*A.*dxdxvx + 7.*dyvy.*px + 5.*A.*dyvy.*px - 13.*dxvx.*px - 11.*A.*dxvx.*px + 5.*dxvy.*py + 7.*A.*dxvy.*py + 2.*dtdtvx + 2.*A.*dtdtvx)) - (-1 + vy.^2).*(dyvy.*dxvy + 3.*A.*dyvy.*dxvy + dxvx.*dxvy - A.*dxvx.*dxvy + 4.*px.*py + 4.*A.*px.*py + dyvx.*((1 - A + 8.*(1 + A).*vy.^2).*dyvy - (1 + 3.*A).*(-1 + vy.^2).*dxvx + vy.*(7 + 5.*A + 2.*(1 + A).*vy.^2).*py) + 2.*dypx + 2.*A.*dypx - 2.*(1 + A).*vy.^4.*dypx + 2.*dxpy + 2.*A.*dxpy + vy.^2.*(4.*(1 + A).*dyvy.*dxvy + (-1 + A).*dxvx.*dxvy + 5.*px.*py + 3.*A.*px.*py - 2.*dxpy - 2.*A.*dxpy) - vy.^3.*(2.*(1 + A).*dydyvx + 2.*(1 + A).*dxdyvy - 2.*dyvy.*px - 2.*A.*dyvy.*px + dxvx.*px + 3.*A.*dxvx.*px + 2.*dtdtvx + 2.*A.*dtdtvx) + vy.*(2.*(1 + A).*dydyvx + 2.*(1 + A).*dxdyvy + 7.*dyvy.*px + 5.*A.*dyvy.*px + dxvx.*px + 3.*A.*dxvx.*px + 5.*dxvy.*py + 7.*A.*dxvy.*py + 2.*dtdtvx + 2.*A.*dtdtvx)) + vx.^5.*(2.*(1 + A).*dxdyvx + 2.*(1 + A).*vy.^2.*(dydyvy - dxdyvx - dxdxvy) + 2.*dxdxvy + 2.*A.*dxdxvy - 2.*dxvy.*px - 2.*A.*dxvy.*px + dyvy.*py + 3.*A.*dyvy.*py - 2.*dxvx.*py - 2.*A.*dxvx.*py + vy.*((1 + A).*dyvy.^2 - 2.*(1 + A).*dyvy.*dxvx - 2.*((1 + A).*dyvx.*dxvy + (1 + A).*dxvy.^2 - A.*py.^2 - 2.*dypy - 2.*A.*dypy)) + 2.*dtdtvy + 2.*A.*dtdtvy) + vx.*(2.*dxdyvx + 2.*A.*dxdyvx + 2.*(1 + A).*vy.^6.*dxdyvx + 2.*dxdxvy + 2.*A.*dxdxvy + 5.*dyvx.*px + 7.*A.*dyvx.*px + 7.*dxvy.*px + 5.*A.*dxvy.*px + dyvy.*py + 3.*A.*dyvy.*py + 7.*dxvx.*py + 5.*A.*dxvx.*py - vy.^4.*(2.*(1 + A).*dydyvy + 2.*(1 + A).*dxdyvx - 2.*dxdxvy - 2.*A.*dxdxvy + 9.*dyvx.*px + 7.*A.*dyvx.*px + 5.*dxvy.*px + 7.*A.*dxvy.*px + 5.*dxvx.*py + 7.*A.*dxvx.*py) + vy.^5.*(-2.*(1 + A).*dyvx.^2 - 2.*(1 + A).*dyvy.*dxvx + dxvx.^2 + A.*dxvx.^2 - 2.*(1 + A).*dyvx.*dxvy + 2.*A.*px.^2 + 4.*dxpx + 4.*A.*dxpx) + vy.*(5.*(1 + A).*dyvx.^2 + (1 + A).*dyvy.^2 + 10.*(1 + A).*dyvy.*dxvx + dxvx.^2 + A.*dxvx.^2 + 10.*(1 + A).*dyvx.*dxvy + 5.*dxvy.^2 + 5.*A.*dxvy.^2 + 7.*px.^2 + 9.*A.*px.^2 + 7.*py.^2 + 9.*A.*py.^2 + 4.*dypy + 4.*A.*dypy + 4.*dxpx + 4.*A.*dxpx) - vy.^3.*(3.*(1 + A).*dyvx.^2 - 6.*(1 + A).*dyvy.^2 + 8.*(1 + A).*dyvy.*dxvx + 2.*dxvx.^2 + 2.*A.*dxvx.^2 + 8.*(1 + A).*dyvx.*dxvy + 5.*dxvy.^2 + 5.*A.*dxvy.^2 + 7.*px.^2 + 11.*A.*px.^2 + 2.*A.*py.^2 + 4.*dypy + 4.*A.*dypy + 8.*dxpx + 8.*A.*dxpx) + 2.*dtdtvy + 2.*A.*dtdtvy + vy.^2.*(2.*(1 + A).*dydyvy - 2.*(1 + A).*dxdyvx - 4.*dxdxvy - 4.*A.*dxdxvy + 4.*dyvx.*px - 2.*dxvy.*px + 2.*A.*dxvy.*px + 13.*dyvy.*py + 11.*A.*dyvy.*py - 2.*dxvx.*py + 2.*A.*dxvx.*py - 2.*dtdtvy - 2.*A.*dtdtvy)) + vx.^3.*(-4.*dxdyvx - 4.*A.*dxdyvx + 2.*(1 + A).*vy.^4.*(dydyvy - dxdxvy) - 4.*dxdxvy - 4.*A.*dxdxvy - 5.*dyvx.*px - 7.*A.*dyvx.*px - 5.*dxvy.*px - 3.*A.*dxvy.*px - 2.*dyvy.*py - 6.*A.*dyvy.*py - 5.*dxvx.*py - 3.*A.*dxvx.*py + vy.^3.*(5.*(1 + A).*dyvx.^2 - 6.*(1 + A).*dyvy.^2 + 10.*(1 + A).*dyvy.*dxvx - 6.*dxvx.^2 - 6.*A.*dxvx.^2 + 10.*(1 + A).*dyvx.*dxvy + 5.*dxvy.^2 + 5.*A.*dxvy.^2 + 2.*A.*px.^2 + 2.*A.*py.^2 + 4.*dypy + 4.*A.*dypy + 4.*dxpx + 4.*A.*dxpx) - vy.*(5.*(1 + A).*dyvx.^2 + 2.*(1 + A).*dyvy.^2 + 8.*(1 + A).*dyvy.*dxvx - 6.*dxvx.^2 - 6.*A.*dxvx.^2 + 8.*(1 + A).*dyvx.*dxvy + 3.*dxvy.^2 + 3.*A.*dxvy.^2 + 2.*A.*px.^2 + 7.*py.^2 + 11.*A.*py.^2 + 8.*dypy + 8.*A.*dypy + 4.*dxpx + 4.*A.*dxpx) - 4.*dtdtvy - 4.*A.*dtdtvy + vy.^2.*(-4.*(1 + A).*dydyvy + 4.*(1 + A).*dxdyvx + 6.*dxdxvy + 6.*A.*dxdxvy + 5.*dyvx.*px + 7.*A.*dyvx.*px + 7.*dxvy.*px + 5.*A.*dxvy.*px - 13.*dyvy.*py - 11.*A.*dyvy.*py + 7.*dxvx.*py + 5.*A.*dxvx.*py + 2.*dtdtvy + 2.*A.*dtdtvy)))))

  B = -((pi.*T.*(6.*sqrt(1 - vx.^2 - vy.^2).*(-1 + vy.^2).*(-A + A.*vx.^2 + (1 + A).*vy.^2).*dyvx.^2 + 8.*pi.*T.*(-1 + vx.^2 + vy.^2).^2.*((-1 + vx.^2).*dyvy - (-1 + vy.^2).*dxvx + vx.*px - vx.*vy.^2.*px - vy.*py + vx.^2.*vy.*py) - 3.*sqrt(1 - vx.^2 - vy.^2).*dyvx.*(2.*(1 + A).*vx.^4.*dxvy - 2.*vy.*(-1 + vy.^2).*((1 + A).*vy.*dxvy + 2.*px) - vx.^2.*(2.*(1 + A).*dxvy + (5 + 7.*A).*vy.*(-1 + vy.^2).*px) + vx.^3.*((5 + 7.*A).*vy.*dyvy + 2.*(1 + A).*py + (3 + 5.*A).*vy.^2.*py) - vx.*((9 + 7.*A).*vy.^3.*dxvx + vy.*((5 + 7.*A).*dyvy - (9 + 7.*A).*dxvx) + 2.*(1 + A).*py + (1 + 3.*A).*vy.^2.*py + 2.*(1 + A).*vy.^4.*py)) + 3.*sqrt(1 - vx.^2 - vy.^2).*((1 + A).*(-1 + vx.^2).*(-1 + vx.^2 - 6.*vy.^2).*dyvy.^2 - dxvx.^2 - A.*dxvx.^2 - 6.*vx.^2.*dxvx.^2 - 6.*A.*vx.^2.*dxvx.^2 - 2.*A.*dxvy.^2 + 2.*vx.^2.*dxvy.^2 + 4.*A.*vx.^2.*dxvy.^2 - 2.*vx.^4.*dxvy.^2 - 2.*A.*vx.^4.*dxvy.^2 + 2.*vx.*dxdyvy + 2.*A.*vx.*dxdyvy - 4.*vx.^3.*dxdyvy - 4.*A.*vx.^3.*dxdyvy + 2.*vx.^5.*dxdyvy + 2.*A.*vx.^5.*dxdyvy - 2.*vx.*dxdxvx - 2.*A.*vx.*dxdxvx + 2.*vx.^3.*dxdxvx + 2.*A.*vx.^3.*dxdxvx - 10.*vx.*dxvx.*px - 10.*A.*vx.*dxvx.*px - 4.*vx.^3.*dxvx.*px - 4.*A.*vx.^3.*dxvx.*px - 2.*px.^2 - 2.*A.*px.^2 - 5.*vx.^2.*px.^2 - 5.*A.*vx.^2.*px.^2 + 4.*vx.*dxvy.*py - 4.*vx.^3.*dxvy.*py + 2.*py.^2 + 2.*A.*py.^2 - 2.*vx.^2.*py.^2 - 4.*A.*vx.^2.*py.^2 + 2.*A.*vx.^4.*py.^2 + dyvy.*((-vx.^3).*((9 + 7.*A).*vy.*dxvy + 2.*(1 + A).*px + (3 + 5.*A).*vy.^2.*px) + vx.*((9 + 7.*A).*vy.*dxvy + 2.*(1 + A).*px + (1 + 3.*A).*vy.^2.*px + 2.*(1 + A).*vy.^4.*px) + vx.^4.*(-2.*(1 + A).*dxvx + (1 + 3.*A).*vy.*py) + 2.*(1 + A).*vy.*((-vy).*dxvx + vy.^3.*dxvx + 5.*py + 2.*vy.^2.*py) + vx.^2.*(2.*(1 + A).*dxvx - vy.*(11 + 13.*A + 4.*(1 + A).*vy.^2).*py)) - 2.*(1 + A).*vy.^5.*(dxdyvx + vx.*dypx) + 2.*dypy + 2.*A.*dypy - 4.*vx.^2.*dypy - 4.*A.*vx.^2.*dypy + 2.*vx.^4.*dypy + 2.*A.*vx.^4.*dypy - 2.*dxpx - 2.*A.*dxpx + 2.*vx.^4.*dxpx + 2.*A.*vx.^4.*dxpx - 2.*vx.*dtdtvx - 2.*A.*vx.*dtdtvx + 2.*vx.^3.*dtdtvx + 2.*A.*vx.^3.*dtdtvx + vy.^2.*(2.*(1 + A).*(1 + 3.*vx.^2).*dxvx.^2 - 2.*A.*(-1 + vx.^2).*dxvy.^2 - 2.*vx.*dxdyvy - 2.*A.*vx.*dxdyvy + 2.*vx.^3.*dxdyvy + 2.*A.*vx.^3.*dxdyvy + 4.*vx.*dxdxvx + 4.*A.*vx.*dxdxvx - 2.*vx.^3.*dxdxvx - 2.*A.*vx.^3.*dxdxvx + vx.*(11 + 13.*A + 4.*(1 + A).*vx.^2).*dxvx.*px + 2.*px.^2 + 4.*A.*px.^2 + 5.*vx.^2.*px.^2 + 5.*A.*vx.^2.*px.^2 - (5 + 7.*A).*vx.*(-1 + vx.^2).*dxvy.*py + 5.*py.^2 + 5.*A.*py.^2 - 5.*vx.^2.*py.^2 - 5.*A.*vx.^2.*py.^2 - 2.*vx.^2.*dypy - 2.*A.*vx.^2.*dypy + 2.*vx.^4.*dypy + 2.*A.*vx.^4.*dypy + 4.*dxpx + 4.*A.*dxpx + 2.*vx.^2.*dxpx + 2.*A.*vx.^2.*dxpx - 2.*vx.^4.*dxpx - 2.*A.*vx.^4.*dxpx + 4.*vx.*dtdtvx + 4.*A.*vx.*dtdtvx - 2.*vx.^3.*dtdtvx - 2.*A.*vx.^3.*dtdtvx) - vy.^4.*((1 + A).*dxvx.^2 + (1 + 3.*A).*vx.*dxvx.*px + 2.*(A.*px.^2 - (1 + A).*vx.^2.*(dypy - dxpx) + (1 + A).*(dypy + dxpx) + (1 + A).*vx.*(dxdxvx + dtdtvx))) + vy.*(2.*(1 + A).*(-1 + vx.^2).^2.*dydyvy - vx.^3.*((5 + 7.*A).*px.*py - 2.*(1 + A).*(dypx - 2.*dxpy)) - vx.*((5 + 7.*A).*dxvx.*dxvy + 2.*(1 + A).*(dypx - dxpy)) + 2.*(1 + A).*vx.^5.*dxpy - 2.*(1 + A).*vx.^4.*(dxvy.*px + dxvx.*py - dtdtvy) - 2.*(1 + A).*(dxdyvx + dxvy.*px + dxvx.*py - dtdtvy) + vx.^2.*(2.*(1 + A).*dxdyvx - (1 + 3.*A).*dxvy.*px - dxvx.*py - 3.*A.*dxvx.*py - 4.*dtdtvy - 4.*A.*dtdtvy)) + vy.^3.*(2.*(1 + A).*(-1 + vx.^2).*dydyvy + vx.*((5 + 7.*A).*dxvx.*dxvy + (5 + 7.*A).*px.*py + 2.*(1 + A).*(2.*dypx - dxpy)) - 2.*(1 + A).*vx.^3.*(dypx - dxpy) + 2.*(1 + A).*(2.*dxdyvx + dxvy.*px + dxvx.*py - dtdtvy) + vx.^2.*(-2.*(1 + A).*dxdyvx + (3 + 5.*A).*dxvy.*px + 3.*dxvx.*py + 5.*A.*dxvx.*py + 2.*dtdtvy + 2.*A.*dtdtvy)))))./ (9.*(-1 + vx).*(1 + vx).*(-1 + vy).*(1 + vy).*(1 - vx.^2 - vy.^2).^(7./2)));


  %  end
  %end


  dxPi = dx(Pi);
  dyPi = dy(Pi);
  dxB = dx(B);
  dyB = dy(B);

  toc


  if (ZERO_HYDRO)
    r = sumT;
    return;
  end


  for t = 1:TIME_STEPS
    disp('time step ');
    disp(t);


% dtpx  = (1./(3.*(1 + A).*(vx.^2 + vy.^2))).*(4.*pi.* T.* (1 - vx.^2 - vy.^2).*(vy.* dyvx - vx.^2.*vy.*dyvx - vx.*dyvy + vx.^3.*dyvy + vx.*dxvx + vx.*vy.^2.*dxvx + vy.*dxvy - vx.^2.*vy.*dxvy + vx.^2.*px + vy.^2.*px)) - (-3.*B.*vx + 12.*B.*vx.^3 - 18.*B.*vx.^5 + 12.*B.*vx.^7 - 3.*B.*vx.^9 + 12.*B.*vx.*vy.^2 - 39.*B.*vx.^3.*vy.^2 + 45.*B.*vx.^5.*vy.^2 - 21.*B.*vx.^7.*vy.^2 + 3.*B.*vx.^9.*vy.^2 - 18.*B.*vx.*vy.^4 + 45.*B.*vx.^3.*vy.^4 - 36.*B.*vx.^5.*vy.^4 + 9.*B.*vx.^7.*vy.^4 + 12.*B.*vx.*vy.^6 - 21.*B.*vx.^3.*vy.^6 + 9.*B.*vx.^5.*vy.^6 - 3.*B.*vx.*vy.^8 + 3.*B.*vx.^3.*vy.^8 - 3.*vy.*Pi + 12.*vx.^2.*vy.*Pi - 18.*vx.^4.*vy.*Pi + 12.*vx.^6.*vy.*Pi - 3.*vx.^8.*vy.*Pi + 12.*vy.^3.*Pi - 36.*vx.^2.*vy.^3.*Pi + 36.*vx.^4.*vy.^3.*Pi - 12.*vx.^6.*vy.^3.*Pi - 18.*vy.^5.*Pi + 36.*vx.^2.*vy.^5.*Pi - 18.*vx.^4.*vy.^5.*Pi + 12.*vy.^7.*Pi - 12.*vx.^2.*vy.^7.*Pi - 3.*vy.^9.*Pi + 2.*A.*pi.*T.*vx.* dyvx.^2 - 2.*A.*pi.*T.*vx.^3.* dyvx.^2 + 7.*pi.*T.*vx.*vy.^2.* dyvx.^2 + 5.*A.*pi.*T.*vx.*vy.^2.* dyvx.^2 - 5.*pi.*T.*vx.^3.*vy.^2.* dyvx.^2 - 3.*A.*pi.*T.*vx.^3.*vy.^2.* dyvx.^2 - 5.*pi.*T.*vx.*vy.^4.* dyvx.^2 - 5.*A.*pi.*T.*vx.*vy.^4.* dyvx.^2 + 5.*pi.*T.*vx.^3.*vy.^4.* dyvx.^2 + 5.*A.*pi.*T.*vx.^3.*vy.^4.* dyvx.^2 - 2.*pi.*T.*vx.*vy.^6.* dyvx.^2 - 2.*A.*pi.*T.*vx.*vy.^6.* dyvx.^2 + pi.*T.*vy.*dyvx.* dyvy + 3.*A.*pi.*T.*vy.*dyvx.* dyvy - 7.*pi.*T.*vx.^2.*vy.* dyvx.* dyvy - 9.*A.*pi.*T.*vx.^2.*vy.* dyvx.* dyvy + 6.*pi.*T.*vx.^4.*vy.* dyvx.* dyvy + 6.*A.*pi.*T.*vx.^4.*vy.* dyvx.* dyvy + 7.*pi.*T.*vy.^3.*dyvx.* dyvy + 5.*A.*pi.*T.*vy.^3.* dyvx.* dyvy - pi.*T.*vx.^2.*vy.^3.* dyvx.* dyvy + A.*pi.*T.*vx.^2.*vy.^3.* dyvx.* dyvy - 6.*pi.*T.*vx.^4.*vy.^3.* dyvx.* dyvy - 6.*A.*pi.*T.*vx.^4.*vy.^3.* dyvx.* dyvy - 8.*pi.*T.*vy.^5.*dyvx.* dyvy - 8.*A.*pi.*T.*vy.^5.* dyvx.* dyvy + 8.*pi.*T.*vx.^2.*vy.^5.* dyvx.* dyvy + 8.*A.*pi.*T.*vx.^2.*vy.^5.* dyvx.* dyvy - pi.*T.*vx.*dyvy.^2 - A.*pi.*T.*vx.* dyvy.^2 + 2.*pi.*T.*vx.^3.* dyvy.^2 + 2.*A.*pi.*T.*vx.^3.* dyvy.^2 - pi.*T.*vx.^5.* dyvy.^2 - A.*pi.*T.*vx.^5.* dyvy.^2 - 5.*pi.*T.*vx.*vy.^2.* dyvy.^2 - 5.*A.*pi.*T.*vx.*vy.^2.* dyvy.^2 + 4.*pi.*T.*vx.^3.*vy.^2.* dyvy.^2 + 4.*A.*pi.*T.*vx.^3.*vy.^2.* dyvy.^2 + pi.*T.*vx.^5.*vy.^2.* dyvy.^2 + A.*pi.*T.*vx.^5.*vy.^2.* dyvy.^2 + 6.*pi.*T.*vx.*vy.^4.* dyvy.^2 + 6.*A.*pi.*T.*vx.*vy.^4.* dyvy.^2 - 6.*pi.*T.*vx.^3.*vy.^4.* dyvy.^2 - 6.*A.*pi.*T.*vx.^3.*vy.^4.* dyvy.^2 + 2.*pi.*T.*vy.^2.* dydyvx + 2.*A.*pi.*T.*vy.^2.* dydyvx - 4.*pi.*T.*vx.^2.*vy.^2.* dydyvx - 4.*A.*pi.*T.*vx.^2.*vy.^2.* dydyvx + 2.*pi.*T.*vx.^4.*vy.^2.* dydyvx + 2.*A.*pi.*T.*vx.^4.*vy.^2.* dydyvx - 4.*pi.*T.*vy.^4.* dydyvx - 4.*A.*pi.*T.*vy.^4.* dydyvx + 6.*pi.*T.*vx.^2.*vy.^4.* dydyvx + 6.*A.*pi.*T.*vx.^2.*vy.^4.* dydyvx - 2.*pi.*T.*vx.^4.*vy.^4.* dydyvx - 2.*A.*pi.*T.*vx.^4.*vy.^4.* dydyvx + 2.*pi.*T.*vy.^6.* dydyvx + 2.*A.*pi.*T.*vy.^6.* dydyvx - 2.*pi.*T.*vx.^2.*vy.^6.* dydyvx - 2.*A.*pi.*T.*vx.^2.*vy.^6.* dydyvx - 2.*pi.*T.*vx.*vy.* dydyvy - 2.*A.*pi.*T.*vx.*vy.* dydyvy + 4.*pi.*T.*vx.^3.*vy.* dydyvy + 4.*A.*pi.*T.*vx.^3.*vy.* dydyvy - 2.*pi.*T.*vx.^5.*vy.* dydyvy - 2.*A.*pi.*T.*vx.^5.*vy.* dydyvy + 4.*pi.*T.*vx.*vy.^3.* dydyvy + 4.*A.*pi.*T.*vx.*vy.^3.* dydyvy - 6.*pi.*T.*vx.^3.*vy.^3.* dydyvy - 6.*A.*pi.*T.*vx.^3.*vy.^3.* dydyvy + 2.*pi.*T.*vx.^5.*vy.^3.* dydyvy + 2.*A.*pi.*T.*vx.^5.*vy.^3.* dydyvy - 2.*pi.*T.*vx.*vy.^5.* dydyvy - 2.*A.*pi.*T.*vx.*vy.^5.* dydyvy + 2.*pi.*T.*vx.^3.*vy.^5.* dydyvy + 2.*A.*pi.*T.*vx.^3.*vy.^5.* dydyvy + pi.*T.*vy.*dyvx.* dxvx - A.*pi.*T.*vy.*dyvx.* dxvx + 12.*pi.*T.*vx.^2.*vy.* dyvx.* dxvx + 16.*A.*pi.*T.*vx.^2.*vy.* dyvx.* dxvx - 4.*pi.*T.*vx.^4.*vy.* dyvx.* dxvx - 4.*A.*pi.*T.*vx.^4.*vy.* dyvx.* dxvx - 2.*pi.*T.*vy.^3.*dyvx.* dxvx + 2.*A.*pi.*T.*vy.^3.* dyvx.* dxvx - 2.*pi.*T.*vx.^2.*vy.^3.* dyvx.* dxvx - 6.*A.*pi.*T.*vx.^2.*vy.^3.* dyvx.* dxvx + 4.*pi.*T.*vx.^4.*vy.^3.* dyvx.* dxvx + 4.*A.*pi.*T.*vx.^4.*vy.^3.* dyvx.* dxvx + pi.*T.*vy.^5.*dyvx.* dxvx - A.*pi.*T.*vy.^5.*dyvx.* dxvx - 10.*pi.*T.*vx.^2.*vy.^5.* dyvx.* dxvx - 10.*A.*pi.*T.*vx.^2.*vy.^5.* dyvx.* dxvx - 2.*pi.*T.*vx.^3.*dyvy.* dxvx - 2.*A.*pi.*T.*vx.^3.* dyvy.* dxvx + 2.*pi.*T.*vx.^5.*dyvy.* dxvx + 2.*A.*pi.*T.*vx.^5.* dyvy.* dxvx + 12.*pi.*T.*vx.*vy.^2.* dyvy.* dxvx + 12.*A.*pi.*T.*vx.*vy.^2.* dyvy.* dxvx - 8.*pi.*T.*vx.^3.*vy.^2.* dyvy.* dxvx - 8.*A.*pi.*T.*vx.^3.*vy.^2.* dyvy.* dxvx - 2.*pi.*T.*vx.^5.*vy.^2.* dyvy.* dxvx - 2.*A.*pi.*T.*vx.^5.*vy.^2.* dyvy.* dxvx - 10.*pi.*T.*vx.*vy.^4.* dyvy.* dxvx - 10.*A.*pi.*T.*vx.*vy.^4.* dyvy.* dxvx + 10.*pi.*T.*vx.^3.*vy.^4.* dyvy.* dxvx + 10.*A.*pi.*T.*vx.^3.*vy.^4.* dyvy.* dxvx - 2.*pi.*T.*vx.*vy.^6.* dyvy.* dxvx - 2.*A.*pi.*T.*vx.*vy.^6.* dyvy.* dxvx + pi.*T.*vx.*dxvx.^2 + A.*pi.*T.*vx.* dxvx.^2 + 6.*pi.*T.*vx.^3.* dxvx.^2 + 6.*A.*pi.*T.*vx.^3.* dxvx.^2 - pi.*T.*vx.*vy.^2.* dxvx.^2 - A.*pi.*T.*vx.*vy.^2.* dxvx.^2 - pi.*T.*vx.*vy.^4.* dxvx.^2 - A.*pi.*T.*vx.*vy.^4.* dxvx.^2 - 6.*pi.*T.*vx.^3.*vy.^4.* dxvx.^2 - 6.*A.*pi.*T.*vx.^3.*vy.^4.* dxvx.^2 + pi.*T.*vx.*vy.^6.* dxvx.^2 + A.*pi.*T.*vx.*vy.^6.* dxvx.^2 - 2.*pi.*T.*vx.^3.*dyvx.* dxvy - 2.*A.*pi.*T.*vx.^3.* dyvx.* dxvy + 2.*pi.*T.*vx.^5.*dyvx.* dxvy + 2.*A.*pi.*T.*vx.^5.* dyvx.* dxvy + 12.*pi.*T.*vx.*vy.^2.* dyvx.* dxvy + 12.*A.*pi.*T.*vx.*vy.^2.* dyvx.* dxvy - 8.*pi.*T.*vx.^3.*vy.^2.* dyvx.* dxvy - 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270.*B.*vx.^5.*vy.^3.*dxvx - 64.*pi.^3.*T.^3.*vx.^5.*vy.^3.*dxvx + 54.*B.*vx.^7.*vy.^3.*dxvx + 243.*B.*vx.*vy.^5.*dxvx + 64.*pi.^3.*T.^3.*vx.*vy.^5.*dxvx - 432.*B.*vx.^3.*vy.^5.*dxvx - 96.*pi.^3.*T.^3.*vx.^3.*vy.^5.*dxvx + 216.*B.*vx.^5.*vy.^5.*dxvx + 32.*pi.^3.*T.^3.*vx.^5.*vy.^5.*dxvx - 27.*B.*vx.^7.*vy.^5.*dxvx - 135.*B.*vx.*vy.^7.*dxvx - 32.*pi.^3.*T.^3.*vx.*vy.^7.*dxvx + 189.*B.*vx.^3.*vy.^7.*dxvx + 32.*pi.^3.*T.^3.*vx.^3.*vy.^7.*dxvx - 54.*B.*vx.^5.*vy.^7.*dxvx + 27.*B.*vx.*vy.^9.*dxvx - 27.*B.*vx.^3.*vy.^9.*dxvx - 18.*vy.^2.*Pi.*dxvx + 72.*vx.^2.*vy.^2.*Pi.*dxvx - 63.*vx.^4.*vy.^2.*Pi.*dxvx + 9.*vx.^6.*vy.^2.*Pi.*dxvx + 54.*vy.^4.*Pi.*dxvx - 153.*vx.^2.*vy.^4.*Pi.*dxvx + 81.*vx.^4.*vy.^4.*Pi.*dxvx - 9.*vx.^6.*vy.^4.*Pi.*dxvx - 54.*vy.^6.*Pi.*dxvx + 90.*vx.^2.*vy.^6.*Pi.*dxvx - 18.*vx.^4.*vy.^6.*Pi.*dxvx + 18.*vy.^8.*Pi.*dxvx - 9.*vx.^2.*vy.^8.*Pi.*dxvx - 64.*pi.^3.*T.^3.*vx.^2.*dxvy + 128.*pi.^3.*T.^3.*vx.^4.*dxvy - 64.*pi.^3.*T.^3.*vx.^6.*dxvy + 36.*B.*vy.^2.*dxvy - 144.*B.*vx.^2.*vy.^2.*dxvy + 128.*pi.^3.*T.^3.*vx.^2.*vy.^2.*dxvy + 216.*B.*vx.^4.*vy.^2.*dxvy - 224.*pi.^3.*T.^3.*vx.^4.*vy.^2.*dxvy - 144.*B.*vx.^6.*vy.^2.*dxvy + 96.*pi.^3.*T.^3.*vx.^6.*vy.^2.*dxvy + 36.*B.*vx.^8.*vy.^2.*dxvy - 108.*B.*vy.^4.*dxvy + 360.*B.*vx.^2.*vy.^4.*dxvy - 96.*pi.^3.*T.^3.*vx.^2.*vy.^4.*dxvy - 432.*B.*vx.^4.*vy.^4.*dxvy + 128.*pi.^3.*T.^3.*vx.^4.*vy.^4.*dxvy + 216.*B.*vx.^6.*vy.^4.*dxvy - 32.*pi.^3.*T.^3.*vx.^6.*vy.^4.*dxvy - 36.*B.*vx.^8.*vy.^4.*dxvy + 108.*B.*vy.^6.*dxvy - 288.*B.*vx.^2.*vy.^6.*dxvy + 32.*pi.^3.*T.^3.*vx.^2.*vy.^6.*dxvy + 252.*B.*vx.^4.*vy.^6.*dxvy - 32.*pi.^3.*T.^3.*vx.^4.*vy.^6.*dxvy - 72.*B.*vx.^6.*vy.^6.*dxvy - 36.*B.*vy.^8.*dxvy + 72.*B.*vx.^2.*vy.^8.*dxvy - 36.*B.*vx.^4.*vy.^8.*dxvy + 36.*vx.*vy.*Pi.*dxvy - 81.*vx.^3.*vy.*Pi.*dxvy + 54.*vx.^5.*vy.*Pi.*dxvy - 9.*vx.^7.*vy.*Pi.*dxvy - 81.*vx.*vy.^3.*Pi.*dxvy + 144.*vx.^3.*vy.^3.*Pi.*dxvy - 72.*vx.^5.*vy.^3.*Pi.*dxvy + 9.*vx.^7.*vy.^3.*Pi.*dxvy + 54.*vx.*vy.^5.*Pi.*dxvy - 72.*vx.^3.*vy.^5.*Pi.*dxvy + 18.*vx.^5.*vy.^5.*Pi.*dxvy - 9.*vx.*vy.^7.*Pi.*dxvy + 9.*vx.^3.*vy.^7.*Pi.*dxvy - 18.*vx.*dxPi + 54.*vx.^3.*dxPi - 54.*vx.^5.*dxPi + 18.*vx.^7.*dxPi + 54.*vx.*vy.^2.*dxPi - 126.*vx.^3.*vy.^2.*dxPi + 90.*vx.^5.*vy.^2.*dxPi - 18.*vx.^7.*vy.^2.*dxPi - 54.*vx.*vy.^4.*dxPi + 90.*vx.^3.*vy.^4.*dxPi - 36.*vx.^5.*vy.^4.*dxPi + 18.*vx.*vy.^6.*dxPi - 18.*vx.^3.*vy.^6.*dxPi - 18.*B.*vy.*px - 64.*pi.^3.*T.^3.*vy.*px + 90.*B.*vx.^2.*vy.*px + 128.*pi.^3.*T.^3.*vx.^2.*vy.*px - 135.*B.*vx.^4.*vy.*px - 96.*pi.^3.*T.^3.*vx.^4.*vy.*px + 72.*B.*vx.^6.*vy.*px + 32.*pi.^3.*T.^3.*vx.^6.*vy.*px - 9.*B.*vx.^8.*vy.*px + 72.*B.*vy.^3.*px + 128.*pi.^3.*T.^3.*vy.^3.*px - 297.*B.*vx.^2.*vy.^3.*px - 224.*pi.^3.*T.^3.*vx.^2.*vy.^3.*px + 369.*B.*vx.^4.*vy.^3.*px + 128.*pi.^3.*T.^3.*vx.^4.*vy.^3.*px - 162.*B.*vx.^6.*vy.^3.*px - 32.*pi.^3.*T.^3.*vx.^6.*vy.^3.*px + 18.*B.*vx.^8.*vy.^3.*px - 108.*B.*vy.^5.*px - 64.*pi.^3.*T.^3.*vy.^5.*px + 351.*B.*vx.^2.*vy.^5.*px + 96.*pi.^3.*T.^3.*vx.^2.*vy.^5.*px - 342.*B.*vx.^4.*vy.^5.*px - 32.*pi.^3.*T.^3.*vx.^4.*vy.^5.*px + 108.*B.*vx.^6.*vy.^5.*px - 9.*B.*vx.^8.*vy.^5.*px + 72.*B.*vy.^7.*px - 171.*B.*vx.^2.*vy.^7.*px + 117.*B.*vx.^4.*vy.^7.*px - 18.*B.*vx.^6.*vy.^7.*px - 18.*B.*vy.^9.*px + 27.*B.*vx.^2.*vy.^9.*px - 9.*B.*vx.^4.*vy.^9.*px - 18.*vx.*Pi.*px + 54.*vx.^3.*Pi.*px - 54.*vx.^5.*Pi.*px + 18.*vx.^7.*Pi.*px + 72.*vx.*vy.^2.*Pi.*px - 135.*vx.^3.*vy.^2.*Pi.*px + 81.*vx.^5.*vy.^2.*Pi.*px - 18.*vx.^7.*vy.^2.*Pi.*px - 81.*vx.*vy.^4.*Pi.*px + 81.*vx.^3.*vy.^4.*Pi.*px - 27.*vx.^5.*vy.^4.*Pi.*px + 18.*vx.*vy.^6.*Pi.*px + 9.*vx.*vy.^8.*Pi.*px + 18.*B.*vx.*py - 64.*pi.^3.*T.^3.*vx.*py - 72.*B.*vx.^3.*py + 128.*pi.^3.*T.^3.*vx.^3.*py + 108.*B.*vx.^5.*py - 64.*pi.^3.*T.^3.*vx.^5.*py - 72.*B.*vx.^7.*py + 18.*B.*vx.^9.*py - 90.*B.*vx.*vy.^2.*py + 128.*pi.^3.*T.^3.*vx.*vy.^2.*py + 297.*B.*vx.^3.*vy.^2.*py - 224.*pi.^3.*T.^3.*vx.^3.*vy.^2.*py - 351.*B.*vx.^5.*vy.^2.*py + 96.*pi.^3.*T.^3.*vx.^5.*vy.^2.*py + 171.*B.*vx.^7.*vy.^2.*py - 27.*B.*vx.^9.*vy.^2.*py + 135.*B.*vx.*vy.^4.*py - 96.*pi.^3.*T.^3.*vx.*vy.^4.*py - 369.*B.*vx.^3.*vy.^4.*py + 128.*pi.^3.*T.^3.*vx.^3.*vy.^4.*py + 342.*B.*vx.^5.*vy.^4.*py - 32.*pi.^3.*T.^3.*vx.^5.*vy.^4.*py - 117.*B.*vx.^7.*vy.^4.*py + 9.*B.*vx.^9.*vy.^4.*py - 72.*B.*vx.*vy.^6.*py + 32.*pi.^3.*T.^3.*vx.*vy.^6.*py + 162.*B.*vx.^3.*vy.^6.*py - 32.*pi.^3.*T.^3.*vx.^3.*vy.^6.*py - 108.*B.*vx.^5.*vy.^6.*py + 18.*B.*vx.^7.*vy.^6.*py + 9.*B.*vx.*vy.^8.*py - 18.*B.*vx.^3.*vy.^8.*py + 9.*B.*vx.^5.*vy.^8.*py - 18.*vy.*Pi.*py + 72.*vx.^2.*vy.*Pi.*py - 81.*vx.^4.*vy.*Pi.*py + 18.*vx.^6.*vy.*Pi.*py + 9.*vx.^8.*vy.*Pi.*py + 54.*vy.^3.*Pi.*py - 135.*vx.^2.*vy.^3.*Pi.*py + 81.*vx.^4.*vy.^3.*Pi.*py - 54.*vy.^5.*Pi.*py + 81.*vx.^2.*vy.^5.*Pi.*py - 27.*vx.^4.*vy.^5.*Pi.*py + 18.*vy.^7.*Pi.*py - 18.*vx.^2.*vy.^7.*Pi.*py)./(18.*(-1 + vx).*(1 + vx).*(-1 + vy).*(1 + vy).*(-1 + vx.^2 + vy.^2).^2.*(vx.^2 + vy.^2));


    dtT =  -((-64.*pi.^3.*T.^2.*vy.*dyT + 128.*pi.^3.*T.^2.*vx.^2.*vy.*dyT - 64.*pi.^3.*T.^2.*vx.^4.*vy.*dyT + 128.*pi.^3.*T.^2.*vy.^3.*dyT - 192.*pi.^3.*T.^2.*vx.^2.*vy.^3.*dyT + 64.*pi.^3.*T.^2.*vx.^4.*vy.^3.*dyT - 64.*pi.^3.*T.^2.*vy.^5.*dyT + 64.*pi.^3.*T.^2.*vx.^2.*vy.^5.*dyT - 32.*pi.^3.*T.^3.*vx.*vy.*dyvx + 32.*pi.^3.*T.^3.*vx.^3.*vy.*dyvx + 32.*pi.^3.*T.^3.*vx.*vy.^3.*dyvx - 32.*pi.^3.*T.^3.*vx.^3.*vy.^3.*dyvx - 9.*Pi.*dyvx + 18.*vx.^2.*Pi.*dyvx - 9.*vx.^4.*Pi.*dyvx + 18.*vy.^2.*Pi.*dyvx - 27.*vx.^2.*vy.^2.*Pi.*dyvx + 9.*vx.^4.*vy.^2.*Pi.*dyvx - 9.*vy.^4.*Pi.*dyvx + 9.*vx.^2.*vy.^4.*Pi.*dyvx + 9.*B.*dyvy - 32.*pi.^3.*T.^3.*dyvy - 27.*B.*vx.^2.*dyvy + 64.*pi.^3.*T.^3.*vx.^2.*dyvy + 27.*B.*vx.^4.*dyvy - 32.*pi.^3.*T.^3.*vx.^4.*dyvy - 9.*B.*vx.^6.*dyvy - 18.*B.*vy.^2.*dyvy + 32.*pi.^3.*T.^3.*vy.^2.*dyvy + 45.*B.*vx.^2.*vy.^2.*dyvy - 64.*pi.^3.*T.^3.*vx.^2.*vy.^2.*dyvy - 36.*B.*vx.^4.*vy.^2.*dyvy + 32.*pi.^3.*T.^3.*vx.^4.*vy.^2.*dyvy + 9.*B.*vx.^6.*vy.^2.*dyvy + 9.*B.*vy.^4.*dyvy - 18.*B.*vx.^2.*vy.^4.*dyvy + 9.*B.*vx.^4.*vy.^4.*dyvy - 9.*vx.*vy.*Pi.*dyvy + 18.*vx.^3.*vy.*Pi.*dyvy - 9.*vx.^5.*vy.*Pi.*dyvy + 9.*vx.*vy.^3.*Pi.*dyvy - 9.*vx.^3.*vy.^3.*Pi.*dyvy - 64.*pi.^3.*T.^2.*vx.*dxT + 128.*pi.^3.*T.^2.*vx.^3.*dxT - 64.*pi.^3.*T.^2.*vx.^5.*dxT + 128.*pi.^3.*T.^2.*vx.*vy.^2.*dxT - 192.*pi.^3.*T.^2.*vx.^3.*vy.^2.*dxT + 64.*pi.^3.*T.^2.*vx.^5.*vy.^2.*dxT - 64.*pi.^3.*T.^2.*vx.*vy.^4.*dxT + 64.*pi.^3.*T.^2.*vx.^3.*vy.^4.*dxT - 9.*B.*dxvx - 32.*pi.^3.*T.^3.*dxvx + 18.*B.*vx.^2.*dxvx + 32.*pi.^3.*T.^3.*vx.^2.*dxvx - 9.*B.*vx.^4.*dxvx + 27.*B.*vy.^2.*dxvx + 64.*pi.^3.*T.^3.*vy.^2.*dxvx - 45.*B.*vx.^2.*vy.^2.*dxvx - 64.*pi.^3.*T.^3.*vx.^2.*vy.^2.*dxvx + 18.*B.*vx.^4.*vy.^2.*dxvx - 27.*B.*vy.^4.*dxvx - 32.*pi.^3.*T.^3.*vy.^4.*dxvx + 36.*B.*vx.^2.*vy.^4.*dxvx + 32.*pi.^3.*T.^3.*vx.^2.*vy.^4.*dxvx - 9.*B.*vx.^4.*vy.^4.*dxvx + 9.*B.*vy.^6.*dxvx - 9.*B.*vx.^2.*vy.^6.*dxvx - 9.*vx.*vy.*Pi.*dxvx + 9.*vx.^3.*vy.*Pi.*dxvx + 18.*vx.*vy.^3.*Pi.*dxvx - 9.*vx.^3.*vy.^3.*Pi.*dxvx - 9.*vx.*vy.^5.*Pi.*dxvx - 32.*pi.^3.*T.^3.*vx.*vy.*dxvy + 32.*pi.^3.*T.^3.*vx.^3.*vy.*dxvy + 32.*pi.^3.*T.^3.*vx.*vy.^3.*dxvy - 32.*pi.^3.*T.^3.*vx.^3.*vy.^3.*dxvy - 9.*Pi.*dxvy + 18.*vx.^2.*Pi.*dxvy - 9.*vx.^4.*Pi.*dxvy + 18.*vy.^2.*Pi.*dxvy - 27.*vx.^2.*vy.^2.*Pi.*dxvy + 9.*vx.^4.*vy.^2.*Pi.*dxvy - 9.*vy.^4.*Pi.*dxvy + 9.*vx.^2.*vy.^4.*Pi.*dxvy - 9.*B.*vx.*px - 32.*pi.^3.*T.^3.*vx.*px + 18.*B.*vx.^3.*px + 32.*pi.^3.*T.^3.*vx.^3.*px - 9.*B.*vx.^5.*px + 27.*B.*vx.*vy.^2.*px + 32.*pi.^3.*T.^3.*vx.*vy.^2.*px - 45.*B.*vx.^3.*vy.^2.*px - 32.*pi.^3.*T.^3.*vx.^3.*vy.^2.*px + 18.*B.*vx.^5.*vy.^2.*px - 27.*B.*vx.*vy.^4.*px + 36.*B.*vx.^3.*vy.^4.*px - 9.*B.*vx.^5.*vy.^4.*px + 9.*B.*vx.*vy.^6.*px - 9.*B.*vx.^3.*vy.^6.*px - 9.*vy.*Pi.*px + 9.*vx.^2.*vy.*Pi.*px + 18.*vy.^3.*Pi.*px - 9.*vx.^2.*vy.^3.*Pi.*px - 9.*vy.^5.*Pi.*px + 9.*B.*vy.*py - 32.*pi.^3.*T.^3.*vy.*py - 27.*B.*vx.^2.*vy.*py + 32.*pi.^3.*T.^3.*vx.^2.*vy.*py + 27.*B.*vx.^4.*vy.*py - 9.*B.*vx.^6.*vy.*py - 18.*B.*vy.^3.*py + 32.*pi.^3.*T.^3.*vy.^3.*py + 45.*B.*vx.^2.*vy.^3.*py - 32.*pi.^3.*T.^3.*vx.^2.*vy.^3.*py - 36.*B.*vx.^4.*vy.^3.*py + 9.*B.*vx.^6.*vy.^3.*py + 9.*B.*vy.^5.*py - 18.*B.*vx.^2.*vy.^5.*py + 9.*B.*vx.^4.*vy.^5.*py - 9.*vx.*Pi.*py + 18.*vx.^3.*Pi.*py - 9.*vx.^5.*Pi.*py + 9.*vx.*vy.^2.*Pi.*py - 9.*vx.^3.*vy.^2.*Pi.*py)./(64.*pi.^3.*T.^2.*(-1 + vx).*(1 + vx).*(-1 + vy).*(1 + vy).*(-1 + vx.^2 + vy.^2)));





    vx = vx + DT * px;
    vy = vy + DT * py;
    px = px + DT * dtpx;
    py = py + DT * dtpy;
    T  = T  + DT * dtT;
    B  = B  + DT * dtB;
    Pi = Pi + DT * dtPi;

    dxvx = dx(vx);
    dyvx = dy(vx);
    dxvy = dx(vy);
    dyvy = dy(vy);
    dxT = dx(T);
    dyT = dy(T);

    dxdxvx = dxdx(vx);
    dxdyvx = dx(dy(vx));
    dydyvx = dydy(vx);
    dxdxvy = dxdx(vy);
    dxdyvy = dx(dy(vy));
    dydyvy = dydy(vy);
    dxdxT = dxdx(T);
    dxdyT = dx(dy(T));
    dydyT = dydy(T);

    dxpx = dx(px);
    dypx = dy(px);
    dxpy = dx(py);
    dypy = dy(py);

    dxPi = dx(Pi);
    dyPi = dy(Pi);
    dxB = dx(B);
    dyB = dy(B);



    phydro();
    pause(0.003);
  end

















    phydro();
    %pause(0.033);

  function r = phydro()
    frame_active = (mod(t-1,int16(1/DT/SECONDS_PER_LIGHTYEAR/30)) == 0);
    if VIDEO & ~frame_active 
      return
    end
    if VIDEO
      disp(t);
    end
    plotsx = 4;
    plotsy = 3;

    subplot(plotsy, plotsx, 1);
    image((T-TEMP_OFFSET) * 256 * 100 + 128); 
    
    % if (VIDEO) & (frame_active)
    %   frame = getframe;
    %   writeVideo(vt,frame);
    % end

    subplot(plotsy, plotsx, 2);
    image(vx*128 + 128);
    % if (VIDEO) & (frame_active)
    %   frame = getframe;
    %   writeVideo(vvx,frame);
    % end

    subplot(plotsy, plotsx, 3);
    image(vy*128 + 128);
    % if (VIDEO) & (frame_active)
    %   frame = getframe;
    %   writeVideo(vvy,frame);
    % end

    subplot(plotsy, plotsx, 4);
    image(sqrt(vx.*vx + vy.*vy)*128 + 128);
    % if (VIDEO) & (frame_active)
    %   frame = getframe;
    %   writeVideo(vv,frame);
    % end

    subplot(plotsy, plotsx, 5);
    image(px*8 + 128);

    subplot(plotsy, plotsx, 6);
    image(py*8 + 128);

    subplot(plotsy, plotsx, 7);
    image(vx + 100000);
    hold on;
    quiver(gridx, gridy, vy(1:VECTOR_SUBSAMPLING:end,1:VECTOR_SUBSAMPLING:end), vx(1:VECTOR_SUBSAMPLING:end,1:VECTOR_SUBSAMPLING:end));
    hold off;

    subplot(plotsy, plotsx, 9);
    image(Pi*2 + 128);
    subplot(plotsy, plotsx, 10);
    image(B*2 + 128);

    subplot(plotsy, plotsx, 11);
    image(dtpx + 128);
    subplot(plotsy, plotsx, 12);
    image(dtpy + 128);

    truesize(fig);
    if (VIDEO) & (frame_active)
      frame = getframe(gcf);
      writeVideo(vt,frame);
    end

    
    % subplot(plotsy, plotsx, 9);
    % image(dtdtvx*8 + 128);
    % subplot(plotsy, plotsx, 10);
    % image(dtdtvy*8 + 128);
    % subplot(plotsy, plotsx, 11);
    % image(Pi*2 + 128);
    % subplot(plotsy, plotsx, 12);
    % image(B*2 + 128);
    return;

  end

  % [-1 0 1
  function r = dx(arg)
    r = zeros(GRID_SIZE, GRID_SIZE);
    for y = 1:GRID_SIZE
      r(1,y) = - arg(GRID_SIZE,y)/2 + arg(2, y)/2;
    end
    for y = 1:GRID_SIZE
      r(GRID_SIZE,y) =  -arg(GRID_SIZE-1,y)/2 + arg(1, y)/2;
    end
    for x = 2:GRID_SIZE-1
      for y = 1:GRID_SIZE
        r(x, y) = (arg(x+1,y) - arg(x-1,y))/2;   
      end
    end 
    r = r / DX;
  end
  function r = dy(arg)
    r = zeros(GRID_SIZE, GRID_SIZE);
    for x = 1:GRID_SIZE
      r(x,1) = - arg(x,GRID_SIZE)/2 + arg(x,2)/2;
    end
    for x = 1:GRID_SIZE
      r(x,GRID_SIZE) =  -arg(x,GRID_SIZE-1)/2 + arg(x,1)/2;
    end
    for x = 1:GRID_SIZE
      for y = 2:GRID_SIZE-1
        r(x, y) = (arg(x,y+1) - arg(x,y-1))/2;   
      end
    end 
    r = r / DX;
  end
  function r = dx2(arg)
    r = d1 * arg;
  end
  function r = dy2(arg)
    r = arg * (d1.');
  end
  function r = dxdx2(arg)
    r = d2 * arg;
  end
  function r = dydy2(arg)
    r = arg * (d2.');
  end
  function r = dxdx(arg)
    r = zeros(GRID_SIZE, GRID_SIZE);
    for y = 1:GRID_SIZE
      r(1,y) = arg(GRID_SIZE,y) + arg(2, y) - 2*arg(1,y);
    end
    for y = 1:GRID_SIZE
      r(GRID_SIZE,y) =  arg(GRID_SIZE-1,y) + arg(1, y) - 2*arg(GRID_SIZE,y);
    end
    for x = 2:GRID_SIZE-1
      for y = 1:GRID_SIZE
        r(x, y) = arg(x+1,y) + arg(x-1,y) - 2*arg(x,y);
      end
    end 
    r = r / DX / DX;
  end
  function r = dydy(arg)
    r = zeros(GRID_SIZE, GRID_SIZE);
    for x = 1:GRID_SIZE
      r(x,1) = arg(x,GRID_SIZE) + arg(x,2) - 2*arg(x,1);
    end
    for x = 1:GRID_SIZE
      r(x,GRID_SIZE) =  arg(x,GRID_SIZE-1) + arg(x,1) - 2*arg(x,GRID_SIZE);
    end
    for x = 1:GRID_SIZE
      for y = 2:GRID_SIZE-1
        r(x, y) = arg(x,y+1) + arg(x,y-1) - 2*arg(x,y);   
      end
    end 
    r = r / DX / DX;
  end

end


